reserve R for non empty RelStr,
  N for net of R,
  i for Element of N;

theorem  :: 1.10. Propostion (ii), p.104
  for T being complete continuous Scott TopLattice, S being upper Subset of T
  holds S is open iff S is Open
proof
  let T be complete continuous Scott TopLattice, S be upper Subset of T;
  thus S is open implies S is Open
  proof
    assume
A1: S is open;
    let x be Element of T;
    assume x in S;
    then ex y be Element of T st y << x & y in S by A1,Th43;
    hence thesis;
  end;
  thus thesis by Th41;
end;
